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 AM  Vol.10 No.1 , January 2019
Construction of Equivalent Functions in Anisotropic Radon Tomography
Abstract:

We consider a real-valued function on a plane of the form

m(x,y,θ)=A(x,y)+Bc(x,y)cos(2θ)+Bs(x,y)sin(2θ)+Cc(x,y)cos(4θ)Cs(x,y)sin(4θ)

that models anisotropic acoustic slowness (reciprocal velocity) perturbations. This “slowness function” depends on Cartesian coordinates and polar angle θ. The five anisotropic “component functions” A (x,y), Bc(x,y), Bs(x,y), Cc(x,y) and Cs(x,y) are assumed to be real-valued Schwartz functions. The “travel time” function d(u, θ) models the travel time perturbations on an indefinitely long straight-line observation path, where the line is parameterized by perpendicular distance u from the origin and polar angle θ; it is the Radon transform of m ( x, y, θ). We show that: 1) an A can always be found with the same d(u, θ) as an arbitrary (Bc,Bs) and/or an arbitrary (Cc,Cs) ; 2) a (Bc,Bs) can always be found with the same d(u, θ) as an arbitrary A, and furthermore, infinite families of them exist; 3) a (Cc,Cs) can always be found with the same d(u, θ) as an arbitrary A, and furthermore, infinite families of them exist; 4) a (Bc,Bs) can always be found with the same d(u, θ) as an arbitrary (Cc,Cs) , and vice versa; and furthermore, infinite families of them exist; and 5) given an arbitrary isotropic reference slowness function m0(x,y), “null coefficients” (Bc,Bs) can be constructed for which d(u, θ) is identically zero (and similarly for Cc,Cs ). We provide explicit methods of constructing each of these “equivalent functions”.

Cite this paper: Menke, W. (2019) Construction of Equivalent Functions in Anisotropic Radon Tomography. Applied Mathematics, 10, 1-10. doi: 10.4236/am.2019.101001.
References

[1]   Hearn, T.M. (1996) Anisotropic Pn Tomography in the Western United States. Journal of Geophysical Research, 101, 8403-8414.
https://doi.org/10.1029/96JB00114

[2]   Wu, H. and Lees, J.M. (1999) Cartesian Parameterization of Anisotropic Traveltime Tomography. Geophysical Journal International, 137, 64-80.
https://doi.org/10.1046/j.1365-246x.1999.00778.x

[3]   Pei, S., Zhao, J., Sun, Y., Xu, Z., Wang, S., Liu, H., Rowe, C.A., ToksÖz, M.N. and Gao, X. (2007) Upper Mantle Seismic Velocities And anisotropy in China Determined through Pn and Sn Tomography. Journal of Geophysical Research, 112, Article ID: B05312.

[4]   Menke, W., Zha, Y., Webb, S.C. and Blackman, D.K. (2015) Seismic Anisotropy Indicates Ridge-Parallel Asthenospheric Flow beneath the Eastern Lau Spreading Center. Journal of Geophysical Research, 120, 976-992.
https://doi.org/10.1002/2014JB011154

[5]   Eddy, C.L., EkstrÖm, G., Nettles, M. and Gaherty, J.B. (2018) Age Dependence and Anisotropy of Surface-Wave Phase Velocities in the Pacific. Geophysical Journal International, 216, 640-658.
https://doi.org/10.1093/gji/ggy438

[6]   Backus, G.E. (1965) Possible Form of Seismic Anisotropy of the Upper Mantle under the Oceans. Journal of Geophysical Research, 70, 3429-3439.
https://doi.org/10.1029/JZ070i014p03429

[7]   Thomsen, L. (1986) Weak Elastic Anisotropy. Geophysics, 51, 1954-1966.
https://doi.org/10.1190/1.1442051

[8]   Aki, K. and Richards, P.G. (2009) Quantitative Seismology. 2nd Edition, University Science Books, Herndon, Virginia.

[9]   Mochizuki, E. (1997) Nonuniqueness of Two-Dimensional Anisotropic Tomography. Seismological Society of America Bulletin, 87, 261-264.

[10]   Menke, W. (2015) Equivalent Heterogeneity Analysis as a Tool for Understanding the Resolving Power of Anisotropic Travel Time Tomography. Seismological Society of America Bulletin, 105, 719-733.
https://doi.org/10.1785/0120140150

[11]   Reed, M. and Simon. B. (1980) Methods of Modern Mathematical Physics: Functional Analysis I. Revised and Enlarged Edition, Academic Press, San Diego.

[12]   HÖrmander, L. (1990) The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis. 2nd Edition, Springer-Verlag, Berlin.

[13]   Stein, E.M. and Shakarchi, R. (2003) Fourier Analysis: An Introduction. Princeton Lectures in Analysis I. Princeton University Press, Princeton New Jersey.

[14]   Deans, S.R. (1993) The Radon Transform and Some of Its Applications. Revised Edition, Dover, Mineola, New York.

[15]   Bracewell, R. (1999) The Fourier Transform and Its Applications. 3rd Edition, McGraw Hill, New York.

[16]   Smith, M.L. and Dahlen, F.A. (1973) Azimuthal Dependence of Love and Rayleigh-Wave Propagation in a Slightly Anisotropic Medium. Journal of Geophysical Research, 78, 3321-3333.
https://doi.org/10.1029/JB078i017p03321

[17]   Seely, R.T. (1973) Calculus of One and Several Variables. 2nd Edition, Scott Foresman, Glenview.

[18]   Menke, W. (2018) Geophysical Data Analysis: Discrete Inverse Theory. Fourth Edition, Elsevier, Amsterdam.

[19]   Tarantola, A. and Valette, B. (1982) Inverse Problems = Quest for Information. Journal of Geophysics, 50, 159-170.

[20]   Menke, W. and Menke, J. (2016) Environmental Data Analysis with MATLAB. Second Edition, Elsevier, Amsterdam.

[21]   Menke, W. and Eilon, Z. (2015) Relationship between Data Smoothing and the Regularization of Inverse Problems. Pure and Applied Geophysics, 172, 2711-2726.
https://doi.org/10.1007/s00024-015-1059-0

[22]   Levenberg, K. (1944) A Method for the Solution of Certain Non-Linear Problems in Least Squares. Quarterly of Applied Mathematics, 2, 164-168.
https://doi.org/10.1090/qam/10666

 
 
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